Theorem ltrneq3

Description: Two translations agree at any atom not under the fiducial co-atom W iff they are equal. (Contributed by NM, 25-Jul-2013)

	Ref	Expression
Hypotheses	cdlemd.l	⊢ ≤ = ( le ‘ 𝐾 )
	cdlemd.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	cdlemd.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	cdlemd.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
Assertion	ltrneq3	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ↔ 𝐹 = 𝐺 ) )

Proof

Step	Hyp	Ref	Expression
1		cdlemd.l	⊢ ≤ = ( le ‘ 𝐾 )
2		cdlemd.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3		cdlemd.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4		cdlemd.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5		simpl1	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
6		simpl2l	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) → 𝐹 ∈ 𝑇 )
7		simpl2r	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) → 𝐺 ∈ 𝑇 )
8		simpl3	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
9		simpr	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) → ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) )
10	1 2 3 4	cdlemd	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) → 𝐹 = 𝐺 )
11	5 6 7 8 9 10	syl311anc	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) → 𝐹 = 𝐺 )
12		fveq1	⊢ ( 𝐹 = 𝐺 → ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) )
13	12	adantl	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ 𝐹 = 𝐺 ) → ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) )
14	11 13	impbida	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ↔ 𝐹 = 𝐺 ) )